Methods and devices to design and fabricate surfaces on contact lenses and on corneal tissue that correct the eye&#39;s optical aberrations

ABSTRACT

Methods and devices are described that are needed to design and fabricate modified surfaces on contact lenses or on corneal tissue that correct the eye&#39;s optical aberrations beyond defocus and astigmatism. The invention provides the means for: 1) measuring the eye&#39;s optical aberrations either with or without a contact lens in place on the cornea, 2) performing a mathematical analysis on the eye&#39;s optical aberrations in order to design a modified surface shape for the original contact lens or cornea that will correct the optical aberrations, 3) fabricating the aberration-correcting surface on a contact lens by diamond point turning, three dimensional contour cutting, laser ablation, thermal molding, photolithography, thin film deposition, or surface chemistry alteration, and 4) fabricating the aberration-correcting surface on a cornea by laser ablation.

1. BACKGROUND OF THE INVENTION

1.1 Measurements of the Eye's Aberrations

There are several objective optical techniques that have been used tomeasure the wavefront aberrations of the eye. The aberroscope, which isdisclosed by Walsh et al. in the Journal of the Optical Society ofAmerica A, Vol.1, pp. 987-992 (1984), projects a nearly collimated beaminto the eye which is spatially modulated near the pupil by a regulargrid pattern. This beam images onto the retina as a small bright diskwhich is modulated by the dark lines of the grid pattern. Since theeye's pupillary aberrations distort the retinal image of the gridpattern, measurements of the distortions on the retina reveal thepupillary aberrations.

The spatially resolved refractometer, which is disclosed by Webb et al.in Applied Optics, Vol. 31, pp. 3678-3686 (1992), projects a smalldiameter collimated beam through the eye's pupil. Instead of beingspatially modulated by a physical grid as with the aberroscope, thespatially resolved refractometer's beam is raster-scanned across theentire pupil. A sequence of retinal images of the focused light isrecorded with each image associated with a particular location at thepupil. A mapping of the relative locations of these retinal imagesreveals the aberrations across the pupil.

Analyzers of retinal point-spread functions have been disclosed by Artalet al. in the Journal of the Optical Society of America A, Vol. 5, pp.1201-1206 (1988). Analyzers of retinal line-spread functions have beendisclosed by Magnante et al. in Vision Science and Its Applications,Technical Digest Series (Optical Society of America, Washington, D.C.),pp. 76-79 (1997). When used to measure the wavefront aberrations of theeye, these spread function analyzers project a small diameter circularbeam into the eye at the center of the pupil. This beam focuses onto theretina as a tiny source of light. The light from this tiny retinalsource scatters back through the dilated pupil. A small circularaperture (approximately 1 mm diameter) in the imaging section of theanalyzer is located conjugate to the pupil plane. This aperture may betranslated up/down or side/side to sample specific regions in the pupilplane where wavefront aberration measurements are sought. An imaginglens focuses the light through the small aperture onto the imaging planeof a camera. Measurements of the relative locations of the focal spotsfor the various locations of the small aperture characterize thepupillary wavefront aberrations.

The Hartmann-Shack wavefront sensor for ordinary lens or mirror testingwas disclosed originally by Shack et al. in the Journal of the OpticalSociety of America, Vol. 61, p. 656 (1971). This type of wavefrontsensor was adapted to measure the wavefront aberrations of the eye byLiang et al., Journal of the Optical Society of America A, Vol. 11, pp.1949-1957 (1994). The Hartmann-Shack wavefront sensor is similar topoint-spread (or line-spread) function analyzers in that: 1) it projectsa fine point of light onto the retina through a small diameter pupil ofthe eye, and 2) the light which is scattered back from the retinathrough the eye's pupil is imaged onto a camera with a lens that isconjugate to the eye's pupil. However, instead of using a single lenswith a moveable small aperture to image the retinal image onto thecamera, the Hartmann-Shack wavefront sensor utilizes a regulartwo-dimensional array of small lenses (commonly called a microlensarray) which is optically conjugate to the eye's pupil to focus the backscattered light from the retinal image onto the camera. Typicaldiameters of individual microlenses range from 0.1 to 1.0 millimeter.With the Hartmann-Shack wavefront sensor, instead of having a singlespot of light corresponding to a single aperture imaged by the camera,there is an array of focused spots imaged by the camera . . . one spotfor each lens in the microlens array. Furthermore, each imaged spot oflight corresponds to a specific location at the eye's pupil.Measurements of the locations of the array of imaged spots are used toquantify the pupillary aberrations.

Measurements of the wavefront aberrations of the eye to a high degree ofprecision using an improved Hartmann-Shack wavefront sensor aredescribed in 1998 U.S. Pat. No. 5,777,719 to Williams and Liang. What isdescribed in U.S. Pat. No. 5,777,719 improves upon what was describedpreviously by Liang et al. in the Journal of the Optical Society ofAmerica A, Vol. 11, pp. 1949-1957 (1994). Device improvements describedin the Williams and Liang 1998 Patent include: 1) a wavefront correctingdeformable mirror, 2) a method to feedback signals to the deformablemirror to correct the wavefront aberrations, and 3) a polarizer usedwith a polarizing beamsplitter to reduce unwanted stray light fromimpinging on the recording camera.

Although the precision of the resulting wavefront aberrationmeasurements cited by Williams and Liang is impressive, theimplementation of a deformable mirror and a feedback loop is very costlyand is not necessary for achieving the purposes of my invention.

Furthermore, the polarizer with polarizing beamsplitter cited in theWilliams and Liang patent are not necessary for achieving the purposesof my invention, and those devices are replaced in my invention with asingle device called an optical isolator (consisting of a polarizerfused to a quarter-wave plate). The optical isolator achieves the samepurpose as the pair of polarizing devices described by Williams andLiang, namely reducing unwanted stray light.

Finally, a laser is cited as the preferred illumination source in theWilliams and Liang patent. However, a conventional laser is improvedupon in my invention through the use of a diode laser operated belowthreshold. Such a light source is not as coherent as a standard laseroperating above threshold, Images formed with such a non-coherent sourceare less granular (having less “speckle”) than those formed by coherentsources. This improvement results in less noisy granularity in themicrolens images and, thereby, improves the accuracy of the imageprocessing which depends on precisely locating the microlens images.

1.2 Analysis of Hartmann-Shack Wavefront Sensor Data to Characterize theEye's Optical Aberrations

The essential data provided by a Hartmann-Shack wavefront sensormodified to measure the human eye are the directions of the optical raysemerging through the eye's pupil. The method of deriving a mathematicalexpression for the wavefront from this directional ray information isdescribed by Liang et al. in the Journal of the Optical Society ofAmerica A, Vol. 11, pp. 1949-1957 (1994). It is also the method cited in1998 U.S. Pat. No. 5,777,719 to Williams and Liang. First, the wavefrontis expressed as a series of Zernike polynomials with each term weightedinitially by an unknown coefficient. Zernike polynomials are describedin Appendix 2 of “Optical Shop Testing” by D. Malacara (John Wiley andSons, New York, 1978). Next, partial derivatives (in x & y) are thencalculated from the Zernike series expansion. Then, these partialderivative expressions respectively are set equal to the measuredwavefront slopes in the x and y directions obtained from the wavefrontsensor measurements. Finally, the method of least-squares fitting ofpolynomial series to the experimental wavefront slope data is employedwhich results in a matrix expression which, when solved, yields thecoefficients of the Zernike polynomials. Consequently, the wavefront,expressed by the Zernike polynomial series, is completely andnumerically determined numerically at all points in the pupil plane. Theleast-squares fitting method is discussed in Chapter 9, Section 11 of“Mathematics of Physics and Modern Engineering” by Sokolnikoff andRedheffer (McGraw-Hill, New York, 1958).

Although the above described methods to calculate the aberratedwavefront of the eye are cited in the Williams and Liang patent, it issignificant to note that there is not any description in their patent ofhow to design an aberration-correcting contact lens or corneal surfacefrom the aberrated wavefront data. These details for designing anaberration-correcting contact lens or corneal surface are not obvious,and require a number of complex mathematical steps. These mathematicaldetails for designing aberration correcting surfaces on contact lensesor on the cornea itself are described fully in my invention.

Furthermore, Williams and Liang demonstrate that the eye's aberrationscan be corrected by properly modifying the surface of a reflectingmirror. However, they do not demonstrate or provide any description ofhow aberration-correcting surfaces can be designed on refractivesurfaces such as those on contact lenses or on the cornea itself. Myinvention gives a detailed mathematical description of how to designsuch refracting optical surfaces that correct the eye's aberrations.

1.3 Fabrication of Conventional Contact Lenses

Conventional contact lenses with spherical or toroidal surface contoursare made routinely using a method called single point diamond turningwhich utilizes very precise vibration-free lathes. The contact lensblank rotates on a spindle while a diamond point tool, moving along aprecise path, cuts the desired surface contour. The end result is asurface which does not need additional polishing, and exhibits excellentoptical qualities in both figure accuracy and surface finish. Figureaccuracy over the lens surface is better than one wavelength of light.Surface finish, which is reported as rms surface roughness, is betterthan 1 micro-inch. Machines of this type and their use are described byPlummer et al. in the Proceedings of the 8th International PrecisionEngineering Seminar (American Society of Precision Engineering, pp.24-29, 1995).

1.4 Corneal Tissue Ablation to Correct Vision

With the advent of the excimer laser, the means are available forrefractive surgeons to flatten and reshape the surface of the cornea inorder to improve vision. The excimer laser selectively removesmicroscopic layers of corneal tissue allowing light rays to focus moresharply on the retina. In the procedure known as photorefractivekeratectomy (PRK), the laser ablates tissue on the surface of thecornea. In the procedure known as laser in-situ keratomileusis (LASIK),the surgeon first creates a flap on the cornea and then uses the laserto reshape tissue below the corneal surface. Layers of tissue as thin as0.25 microns can be ablated.

With current laser procedures, it is possible only to correct relativelycoarse or low order aberrations of the eye, namely high levels ofnearsightedness, and moderate amounts of farsightedness and astigmatism.With the analytical methods of my invention, which take into account thecorneal shape, and both the low order and higher order aberrations ofthe eye, a modified corneal shape is found which allows all rays fromexternal point objects to focus sharply on the retina. By the meansoffered by my invention, refractive surgery procedures to improve visionwill be improved greatly.

2. SUMMARY OF INVENTION

Conventional spectacles and contact lenses are able to correct thevisual acuity of most people to 20/20 or better. For these individuals,the most significant refractive errors are those caused by the so-calledlowest order optical aberrations, namely defocus, astigmatism and prism.However, there are many people with normal retinal function and clearocular media who cannot be refracted to 20/20 acuity with conventionalophthalmic lenses because their corneal surfaces are extraordinarilyirregular. In this group are patients with severe irregular astigmatism,keratoconus, corneal dystrophies, post penetrating keratoplasty,scarring from ulcerative keratitis, corneal trauma with and withoutsurgical repair, and sub-optimal outcome following refractive surgery.The eyes of these people have abnormal amounts of higher order orirregular optical aberrations. An objective of the invention is toimprove the vision of these patients. A further objective is to providethe best vision possible to individuals with ordinary near andfarsightedness and astigmatism. To achieve these objectives, methods anddevices are described that are used to design and fabricate modifiedsurfaces on contact lenses or on corneal tissue that correct the eye'soptical aberrations beyond defocus and astigmatism.

The objectives of the invention are accomplished by measuring thewavefront aberrations of a subject's eye (either with or without acontact lens on the cornea) using a device that projects a small pointof light on the retina near the macula, re-images the light scatteredback from the retina that emerges from the pupil onto a microlens array,and records the focal spots formed from this light when it is imaged bya microlens array on the image plane of an electronic camera. The imageformed on the camera is conveyed to a computer. The computer utilizesmethods to determine the coordinates of the focal spots and then tocalculate the wavefront slopes of rays emerging from the subject's eye.

The objectives of the invention are further accomplished by mathematicalmethods which analyze the wavefront slope data as well as the shape ofthe subject's original contact lens or corneal surface in order todesign a modified surface shape for the original contact lens or corneathat corrects the aberrations. The steps in this mathematical methodare: 1) determining the normal vectors to the original contact lens orcorneal surface, 2) from these normal vectors and the wavefront slopedata, determine the partial derivatives of the surface for the modifiedcontact lens or corneal surface that corrects the aberrations, and 3)fitting these partial derivatives of the aberration-correcting surfacewith the corresponding partial derivatives of a polynomial expressionthat best represents the aberration-correcting surface. From thesemethods, a mathematical expression for the aberration-correcting surfaceis obtained.

The objectives of the invention are accomplished finally by providingdevices and methods to fabricate the modified aberration-correctingsurfaces designed by the mathematical methods described. For contactlenses, these fabrication devices and methods include those of diamondpoint micro-machining, laser ablation, thermal molding, photolithographyand etching, thin film deposition, and surface chemistry alteration. Forcorneal tissue resurfacing, these fabrication devices and methods arethose associated with laser ablation as used with photorefractivekeratectomy (PRK) and laser in-situ keratomileusis (LASIK).

The invention may be more clearly understood by reference to thefollowing detailed description of the invention, the appended claims,and the attached drawings.

3. DESCRIPTION OF DRAWINGS

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4. DETAILED DESCRIPTION OF THE INVENTION

4.1 Wavefront Sensor for Measuring the Eye's Optical Aberrations

A schematic drawing of a wavefront sensor which has been modified tomeasure the eye's optical aberrations is shown in FIG. 1. The design andoperating principles of the subassemblies of the wavefront sensor areexplained in detail below.

4.1.1 Projection System

In order to reduce bothersome “speckle” from coherence effects fromconventional laser sources, non-coherent optical sources are preferred.Thus, source 1 can be anyone of the following: laser diode operatingbelow threshold, light emitting diode, arc source, or incandescentfilament lamp. The source beam is deflected by fold mirror 2, andfocused by microscope objective lens 3 onto a small pinhole aperture 4having a diameter typically in the range from 5 to 15 microns. Anotherlens 5 collimates the beam which next passes through polarizer 6 andthen through aperture stop 7. Typically this stop restricts the beamdiameter to about 2 mm or less. Following the stop is an electronicshutter 8 used to control the light exposure to the patient duringmeasurement to about 1/10 sec. Beamsplitter 9 deviates the collimatedbeam by 90 degrees. The beam then passes through optical isolator 10which consists of a quarter-wave plate and polarizer. Lens 11 forms afocused point image at the center of field stop 12 which the subjectviews through focusing eyepiece 13. The subject's eye 15 then images thelight to a point spot on the retina 16. The field stop 12 and theoptical isolator 10 both serve the important function of blockingbothersome corneal specular reflections and instrument lens reflectionsfrom reaching photo-electronic imaging device 20 such as a vidiconcamera, a charge-coupled device or CCD camera, or a charge-injectiondevice camera.

4.1.2 Camera System

Since the retina acts as a diffuse reflector, some of the light from theretinal point image 16 is reflected back out of the eye 15 through thepupil and cornea 14. The beam emerging from the eye has its polarizationrandomized due to passage through the eye's birefringent cornea 14 andlens 17 as well due to scattering by the diffuse retina 16. Passing nowin reverse direction through lens 13, field stop 12, lens 11, andoptical isolator 10, the beam, which is now aberrated by the eye'soptics, is incident from the right side onto beamsplitter 9 whichtransmits about half its intensity straight through to relay lens pair18. Collectively, lenses 13, 11, and relay lens pair 18 serve tore-image subject's pupil 14 onto the plane of the microlens array 19with unit magnification. In this way the aberrant wavefront emergingfrom the subject's pupil 14 is mapped exactly onto the microlens array19. As shown in FIG. 2, each tiny lens 23 of the array images a portionof the aberrated wavefront 26 onto its focal plane 21 at or near itsaxis 25. The regular array of microlenses 19 produces a correspondingarray of focal spots 24. Deviations of focal spots from the respectivemicrolens axes 25 manifest the wavefront slope error over the entiresurface of the microlens array (and, correspondingly, the subject'spupil). The input image plane of CCD camera 21 coincides with the focalplane 21 of the microlens array. Photo-electronic imaging device 20interfaces with a computer equipped with a “frame-grabber” board (notshown) controlled by appropriate software. An image of the array offocal spots 24 formed by the microlens array 19 appears “live” on thecomputer's monitor, and is ready for “capture” by the computer when ameasurement is taken.

4.1.2.1 Wavefront Slope Measurement

The nature of the wavefront slope measurement is explained now ingreater detail. If a perfect plane wave is incident normally onto aperfect lens which has a small aperture near its surface, the rayspassing through the aperture will be focused by the lens to the lens'sfocal point located on the lens's axis. Regardless of the location ofthe small aperture with respect to the lens surface, the imaged pointwill be at the same location. On the other hand, suppose the wavefrontis imperfect (i.e. individual rays randomly directed and not parallel tothe perfect lens's optical axis). The rays going through the smallaperture now form an image at the lens's focal plane that is displacedfrom the lens's focal point. The displacement of the centroid of theimaged spot (between the perfect and imperfect wave measurements)divided by the distance between the lens and its focal plane (i.e. thefocal length of the lens) equals the angular slope (measured in radians)of the wavefront at the location of the small aperture. Repeating thistype of measurement for many locations of the small aperture over thelens surface fully characterizes the wavefront slope errors at thevarious measurement locations on the lens surface. In a wavefront sensorsuch as shown in FIG. 1, the moveable small aperture with single largediffraction-limited lens is replaced with an array of identicalmicrolenses 19 where each one samples the wavefront at a particularlocation. Details of the microlens array and imaging camera used in thewavefront sensor are shown in FIG. 2. At the focal plane 21 of themicrolens array 19 is the imaging surface of a photo-electronic imagingdevice 20 which records the locations of the focal spots 24 for all themicrolenses in the array. The displacement of each focal spot 24 fromthe optical axis of its associated microlens 25 divided by the focallength of the microlens array equals the slope of the wavefront at themicrolens's location. The locations of the optical axes of theindividual microlenses are determined by a calibration procedure thatinvolves doing a measurement when an unaberrated wavefront 27 (i.e.uniform plane wave) is incident perpendicularly onto microlens array 19.Such an unaberrated wavefront is obtained by replacing the human eye 15shown in FIG. 1 with a diffraction-limited lens and imaging screenplaced at the diffraction-limited lens's focal plane.

4.1.3 Pupil Alignment System

In FIG. 1, field stop 12 is a small hole bored in the direction of theinstrument's main optical axis through a mirrored planar substrateoriented at 45 degrees. The field stop is at the focal plane of thesubject's eyepiece 13. Another eyepiece 22 (called the examiner'seyepiece) is oriented at 90 degrees to the instrument's main opticalaxis so that the examiner can view the subject's pupil 14 by the meansprovided by lens 13 and the mirrored planar substrate of field stop 12.Optionally, a small video camera can be attached to examiner's eyepiece22 so that a video image of the subject's pupil 14 can be viewed on amonitor. By either of these means, the examiner can accurately positionthe subject's eye 15 so that the entering beam is accurately centeredwith respect to the subject's pupil 14. The positioning of the subject'seye with respect to the instrument beam is controlled by a mechanism(not shown) consisting of an x-y-z translation stage that moves a chinand head rest used by the subject.

4.1.4 Data Acquisition and Processing

The subject is asked to look through eyepiece 13 at the point of lightformed within field stop 12, while the examiner adjusts the location ofsubject's eye 15 so that the beam passes through the center of the pupil14. Prior to taking a measurement, the examiner focuses eyepiece 13trying to achieve the brightest and best-focused image of the array offocal spots seen on the computer monitor. When the instrument is alignedwith respect to the patient's eye and a best-focus image is obtained,the operator presses a key which commands the computer to acquire animage of the array of spots. During a measurement session, as many asten successive images may be acquired for subsequent averaging toimprove the signal/noise ratio of the data. The image analysis programcarries out the following steps: 1) subtracts “background” light fromthe image, 2) determines the x & y coordinates (in pixels) of thecentroid for each of the focal spots, 3) subtracts the x & y pixelcoordinate values from a corresponding set of reference values (obtainedfrom a calibration with a diffraction-limited reference lens), 4)multiplies the difference values (in pixel units) by a calibrationfactor which gives for each location in the pupil the components in thex and y directions of the wavefront slope error measured in radians. Thecomponents of the wavefront slope error, labeled Bx and By in thefollowing sections, are the essential measurement data of the wavefrontsensor.

4.2 Design of Aberration-Correcting Lens from Analysis of WavefrontSensor Data

A purpose of the invention is to design modifications to an initiallyknown lens surface, described by z(x,y), which will correct the eye'soptical aberrations measured with wavefront sensors through thatsurface. In this section, the mathematical equations needed for thistask, which leads to a new lens surface described by z′(x,y), arederived. The equations also are applied in an illustrative example. Themathematical formalism in this section is divided into the followingparts: 1) description of original optical surface, 2) obtaining thedirectional derivatives of z′(x,y) from the wavefront sensor data, 3)obtaining a polynomial expansion representing z′(x,y) using the methodof least squares, 4) illustrative example leading to z′(x,y), and 5)demonstration that z′(x,y) corrects the original aberrations. Thefollowing is a guide to the mathematical symbols (whether primed ornot): a) x & y (and X & Y) are coordinates, b) n, R, Brms, a_(j) & b_(j)are scalars, c) z, δz/δx, δz/δy, MAG, α, β, λ, g_(j), ERROR and CUT arefunctions of x & y, d) N, T, A, B, grad g_(j) and grad z arethree-dimensional vector functions of x & y, e) a & b are generalizedvectors, and f) M and M⁻are generalized square matrices.

4.2.1 Original Optical Surface

By “original optical surface” is meant the anterior surface of either acontact lens placed on the cornea or, in the absence of a contact lens,the cornea itself. The index of refraction associated with the surface'sdenser side is n. The original optical surface is represented by z(x,y)which is the distance of the surface from various points in the x-yplane of the pupil. The unit vector perpendicular to the optical surface(called the normal vector N) has components in the x, y and z directionsgiven by N=(Nx,Ny,Nz) where: $\begin{matrix}{{N_{x} = {\frac{1}{MAG} \cdot \left\lbrack \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad x} \right\rbrack}},{{Ny} = {{{\frac{1}{MAG} \cdot {\left\lbrack \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta y} \right\rbrack.{and}}}\quad N_{z}} = \frac{1}{MAG}}}} & (1) \\{{{defining}\quad\ldots\quad{MAG}} = \sqrt{1 + \left\lbrack \frac{\delta\quad{z\left( {x,y} \right)}}{\delta\quad x} \right\rbrack^{2} + \left\lbrack \frac{\delta\quad{z\left( {x,y} \right)}}{\delta\quad y} \right\rbrack^{2}}} & (2)\end{matrix}$

The x & y partial derivatives of the function describing the surfacealso can be expressed in terms of the components of the surface normalby rearranging the terms of Equa. 1. $\begin{matrix}{\frac{\delta\quad{z\left( {x,y} \right)}}{\delta\quad x} = {{{- \left\lbrack \frac{Nx}{Nz} \right\rbrack}\quad{and}\quad\frac{\delta\quad{z\left( {x,y} \right)}}{\delta\quad y}} = {- \left\lbrack \frac{Ny}{Nz} \right\rbrack}}} & (3)\end{matrix}$

4.2.2 Obtaining the Partial Derivatives of the Surface Function whichDescribes the Aberration-Correcting Optical Surface

The light rays, which emanate from the retinal “point source” formed bythe wavefront sensor's projection system, emerge from the eye at theoriginal optical surface which is described by z(x,y). Rays striking thesurface from the denser side are described as A-vectors, and raysleaving the surface into air are described as B-vectors. Both A and Bare vectors of unit length. Refer now to FIG. 3 to express the A and Bvectors in terms of their components along N (surface normal) and T.Note that T is a unit vector tangent to the optical surface at the pointof conjunction of (and coplanar with) rays A and B.A≡cos(α)·N+sin(α)·T   (4)B≡cos(β)·N+sin(β)·T   (5)Next, substitute the expressions for A and B into n·A−B.n·A−B≡(n·cos(α)−cos(β))·N+(n·sin)(α)·sin(β))·T   (6)The second term vanishes due to Snell's Law of Refraction which is:n·sin(α)≡sin(β)   (7)From FIG. 3 and the definition of a vector cross product:sin(β)≡|N·B|  (8)The upright pair of lines in Equa. 8 indicates the magnitude of thevector enclosed by the pair.Combining the results above, find the following equations:$\begin{matrix}{\beta = {a\quad{\sin\left( {{N \times B}} \right)}}} & (9) \\{\alpha = {a\quad{\sin\left\lbrack {\frac{1}{n} \cdot {\sin(\beta)}} \right\rbrack}}} & (10) \\{A = {\frac{1}{n} \cdot \left( {\beta + {\lambda \cdot N}} \right)}} & (11)\end{matrix}$where by definitionλ≡n·cos(α)−cos(β)   (12)

Note that the B-vectors are known quantities representing the wavefrontslopes as measured by the wavefront sensor. Using Equas. 9 through 12,the known quantities (B, N and n) determine the unknown quantities (β,α, λ and A).

Next, solutions for the A-vectors are used to find N′, the unit normalto the aberration-correcting optical surface. This new surface causesthe emerging B′-vectors to lie parallel to the z-axis. In vector form,B′=(0,0,1). The expression for N′ is obtained by rearranging Equa. 11 toconform with this new situation. $\begin{matrix}{N^{\prime} = {\frac{1}{\lambda^{\prime}} \cdot \left( {{n \cdot A} - B^{\prime}} \right)}} & (13) \\{{{where}\quad\ldots\quad\lambda^{\prime}} = {{{n \cdot A} - B^{\prime}}}} & (14)\end{matrix}$Finally the following partial derivatives of the newaberration-correcting optical surface, described by z′(x,y), areobtained in terms of the components of unit vector N′=(Nx′,Ny′,Nz′) byusing Equa. 3 (only adapted to the new surface parameters):$\begin{matrix}{\frac{\delta\quad{z^{\prime}\left( {x,y} \right)}}{\delta\quad x} = {- \left\lbrack \frac{{Nx}^{\prime}}{{Nz}^{\prime}} \right\rbrack}} & (15) \\{\frac{\delta\quad{z^{\prime}\left( {x,y} \right)}}{\delta\quad y} = {- \left\lbrack \frac{{Ny}^{\prime}}{{Nz}^{\prime}} \right\rbrack}} & (16)\end{matrix}$

4.2.3 Obtaining a Polynomial Expression to Represent z′(x,y)

The next problem is to find the new optical surface z′(x,y) from thepartial derivatives expressed by Equa. 15 and 16, which are determinedat discrete coordinate locations in the pupil plane, [x_(k),y_(k)],where the wavefront sensor data are obtained. Begin by expressingz′(x,y) as a polynomial consisting of linearly independent terms,g,(x,y). $\begin{matrix}{{z^{\prime}\left( {x,y} \right)} = {\sum\limits_{j}{\alpha_{j} \cdot {g_{j}\left( {x,y} \right)}}}} & (17)\end{matrix}$The terms, g_(j)(x,y) , can be almost any mathematical functions;however, they generally are restricted to products of powers of x and y,such as x³·y², or sums and differences of such products. There is nonecessity that the functions be orthogonal over the plane of the pupilas are, for example, Zernike polynomials over a circular domain. Thecoefficients, a_(j), are constants which are determined by the methodsderived below. Define “grad”, the gradient, as a mathematical operatorwhich transforms a scalar function, f(x,y) to a vector with componentsin the x and y directions that are, respectively, the partialderivatives of the function with respect to x and y.grad f(x,y)=[δf(x,y)/δx, δf(x,y)/δy]  (18)Applying this operator to both sides of Equa. 17, find $\begin{matrix}{{{grad}\quad{z^{\prime}\left( {x,y} \right)}} = {\sum\limits_{j}{{\alpha_{j} \cdot {grad}}\quad{g_{j}\left( {x,y} \right)}}}} & (19)\end{matrix}$

To find the a-coefficients which provide the “best fit” to the data(i.e. the grad z′ data expressed by Equa. 15 & 16), define the “ERROR”function as the sum of the squares of the differences between the gradz′(x,y) data from Equas. 15 & 16 (i.e. the left side of Equa. 19) andthe estimated value of grad z′(x,y) (i.e. the right side of Equa. 19)for all the measurement points, designated by the index “k”, in the x-ypupil plane. $\begin{matrix}{{ERROR} = {\sum\limits_{k}\left\lbrack {{\sum\limits_{j}{\alpha_{j} \cdot {{grad}\left\lbrack g_{j} \right\rbrack}}} - {{grad}\left( z^{\prime} \right)}} \right\rbrack^{2}}} & (20)\end{matrix}$For brevity sake, the notation for the variables (x,y) has beensuppressed when writing the functions, g_(j)(x,y) and z′(x,y), in Equa.20. This abbreviated notation also is used in the equations whichfollow.

The “method of least squares” determines the a-coefficients byminimizing the ERROR function. This is done by differentiating ERRORwith respect to each element a_(j) and setting each resulting equationto zero: δ(ERROR)/ δa_(j)≡0. There are as many equations resulting fromthis process as there are terms in the expansion for z′(x,y) shown inEqua. 17. The resulting system of linear equations is written:$\begin{matrix}{{\sum\limits_{j}{\left\lbrack {\sum\limits_{k}\left\lbrack {{{grad}\left\lbrack g_{i} \right\rbrack} \cdot {{grad}\left\lbrack g_{j} \right\rbrack}} \right\rbrack} \right\rbrack \cdot \alpha_{j}}} = {\sum\limits_{k}\left\lbrack {{{grad}\left\lbrack g_{i} \right\rbrack} \cdot {{grad}\left( z^{\prime} \right)}} \right\rbrack}} & (21)\end{matrix}$Note that the sums over the index “k” imply a summation over all the(x_(k),y_(k)) coordinates in the pupil plane for the several g-functionsand the grad z′ values. Also, the products indicated in Equa. 21 arevector scalar products or, so-called, “dot products”.

Defining matrix, M, and vector, b, in Equa. 22 & 23 below, Equa. 21appears in the much simpler form which is shown by either Equa. 24 orEqua. 25. Note that M is a symmetric square matrix. $\begin{matrix}{M_{i,j} = {{\sum\limits_{k}{\left\lbrack {{{grad}\left\lbrack g_{i} \right\rbrack} \cdot {{grad}\left\lbrack g_{j} \right\rbrack}} \right\rbrack\quad{so}\quad{that}\quad M_{i,j}}} = M_{j,i}}} & (22) \\{b_{i} = {\sum\limits_{k}\left\lbrack {{{grad}\left\lbrack g_{i} \right\rbrack} \cdot {{grad}\left( z^{\prime} \right)}} \right\rbrack}} & (23)\end{matrix}$Thus, finding the a-coefficients is equivalent to solving the system oflinear equations shown in Equa. 24 where matrix, M, and vector, b, areknown quantities. $\begin{matrix}{{\sum\limits_{j}{M_{i,j} \cdot a_{j}}} = b_{i}} & (24)\end{matrix}$

Using the conventions of standard matrix algebra, Equa. 24 may bewritten:

 M·a≡b   (25)

The solution for vector, a, follows by using standard methods of linearalgebra to obtain the inverse of matrix M which is designated by M⁻¹.Thus, the solution for vector, a, is shown as Equa. 26.a≡M⁻¹·b   (26)

The set of a-coefficients found from Equa. 26, using values of M (Equa.22) and b (Equa. 23), is the final result. With the a-coefficientsdetermined, the explicit “best fit” polynomial expansion for z′(x,y),shown by Equa. 17, is determined.

4.2.4 Illustrative Example

Consider, as an example, when the original optical surface is an acryliccontact lens with an index of refraction n=1.49, and with an anteriorcontour in the shape of a paraboloid described by${z\left( {x,y} \right)} = {- {\left\lbrack {\frac{1}{2 \cdot R} \cdot \left\lbrack {x^{2} + y^{2}} \right\rbrack} \right\rbrack.}}$The radius of curvature at the apex is 7.8 mm which is the constant R.All linear dimensions in this example, such as those described by x, yand z, are understood to be in millimeters.

The partial derivatives of z(x,y) are readily obtained. When thesevalues are substituted into Equas. 1 and 2, the values for thecomponents of the normal vector N as functions of x & y are readilyobtained. $\begin{matrix}\begin{matrix}{{Nx} = \frac{x}{\sqrt{R^{2} + x^{2} + y^{2}}}} \\{{Ny} = \frac{y}{\sqrt{R^{2} + x^{2} + y^{2}}}} \\{{Nz} = \frac{R}{\sqrt{R^{2} + x^{2} + y^{2}}}}\end{matrix} & (27)\end{matrix}$

In this example, measurements with the wavefront sensor are spaced 1 mmapart in the x and y directions within the domain of a circular pupilhaving an 8 mm diameter. There are 49 measurement sites. These data,which are the components in x & y of the emerging B-rays (i.e. Bx & By),are given by the two matrices shown below. The x-direction is to theright, and the y-direction is upward. Note that the values of Bx & Byare multiplied by 1000 which makes the resulting directional units inmilliradians. $\begin{matrix}{1000 \cdot} & (28) \\{\quad{{Bx} = \left\lbrack \quad\begin{matrix}{0} & {0} & {0} & {0} & 0.2 & {0} & {0} & {0} & {0} \\{0} & {0} & {0{.76}} & {0{.57}} & 0.2 & {{- 0}{.17}} & {{- 0}{.36}} & {0} & {0} \\{0} & {0{.86}} & {0{.95}} & {0{.67}} & 0.2 & {{- 0}{.27}} & {{- 0}{.55}} & {{- 0}{.46}} & 0 \\{0} & {0{.95}} & {1{.01}} & {0{.7}} & 0.2 & {{- 0}{.3}} & {{- 0}{.61}} & {{- 0}{.55}} & {0} \\{0{.2}} & {0{.86}} & {0{.95}} & {0{.67}} & 0.2 & {{- 0}{.27}} & {{- 0}{.55}} & {{- 0}{.46}} & {0{.2}} \\{0} & {0{.58}} & {0{.76}} & {0{.58}} & 0.2 & {{- 0}{.18}} & {{- 0}{.36}} & {{- 0}{.18}} & {0} \\{0} & {0{.11}} & {0{.45}} & {0{.42}} & 0.2 & {{- 0}{.02}} & {{- 0}{.05}} & {0{.29}} & {0} \\{0} & {0} & {0{.02}} & {0{.2}} & 0.2 & {0{.2}} & {0{.38}} & {0} & {0} \\{0} & {0} & {0} & {0} & 0.2 & {0} & {0} & {0} & {0}\end{matrix} \right\rbrack}} & \quad \\{1000 \cdot} & (29) \\{\quad{{By} = \left\lbrack \quad\begin{matrix}0 & {0} & {0} & 0 & {{- 2}{.99}} & {0} & {0} & {0} & 0 \\0 & {0} & {{- 2}{.24}} & {- 2.43} & {{- 2}{.5}} & {{- 2}{.43}} & {{- 2}{.24}} & {0} & 0 \\0 & {{- 1}{.34}} & {{- 1}{.5}} & {- 1.59} & {{- 1}{.62}} & {{- 1}{.59}} & {{- 1}{.5}} & {{- 1}{.34}} & 0 \\0 & {{- 0}{.56}} & {{- 0}{.56}} & {- 0.56} & {{- 0}{.56}} & {{- 0}{.56}} & {{- 0}{.56}} & {{- 0}{.56}} & 0 \\0 & {0{.22}} & {0{.38}} & 0.47 & {0{.5}} & {0{.47}} & {0{.38}} & {0{.22}} & 0 \\0 & {\quad{0{.82}}} & {1{.13}} & 1.31 & {1{.38}} & {1{.31}} & {1{.13}} & {0{.82}} & 0 \\0 & {1{.04}} & {1{.51}} & 1.79 & {1{.88}} & {1{.79}} & {1{.51}} & {1{.04}} & 0 \\0 & {0} & {1{.32}} & 1.7 & {1{.82}} & {1{.7}} & {1{.32}} & {0} & 0 \\0 & {0} & {0} & 0 & {1{.02}} & {0} & {0} & {0} & 0\end{matrix}\quad \right\rbrack}} & \quad\end{matrix}$

Bx & By represent the x & y directions of the rays emerging from thecontact lens. The X and Y coordinates of the focal spots resulting fromthese rays, when imaged by the 1 mm spaced microlens array onto thewavefront sensor's measurement plane, are shifted somewhat from theintersection of the regular 1 mm spaced grid lines. The X & Y coordinateshifts are equal to the product of the focal length of the microlensarray and the Bx and By values. In this example f=200 mm. The centroidsof the focal spots of the wavefront sensor pattern are shown in FIG. 4.

A measure of the degree of collimation of all the B-rays is itsroot-mean-square value, labeled Brms. It is found by taking the squareroot of the average of all 49 values of Bx²+By² as shown in Equas. 28 &29. The value is Brms=0.00148 radians.

Continuing with the methods to find the correcting surface, substitutethe known numerical data (i.e. N summarized by Equa. 27, Bx and By givenexplicitly by Equas. 28 and 29, and n=1.49) into Equas. 9 through 12 inorder to find the emerging A-rays. Then, Equas. 13 through 16 are usedto find δz′(x,y)/δx and δz′(x,y)/δy. Next, like Equa. 17, the followingthird order polynomial expression is used to represent z′(x,y).z′(x,y)≡a₁x+a₂·y+a₃·x²+a₄·x·y+a₅·y²+ . . . +a₆·x³+a₇·x²·y+a₈·x·y²+a₉·y³  (30)where g₁≡x g₂≡y g₃≡x² g₄≡x·y g₅≡y⁵ g₆≡x³ g₇≡x²·y g₈≡x·y² g₉≡y³   (31)

Finally, M, b and a are computed from Equas. 22, 23 and 26. The resultsof these calculations appear below. Note that the elements of thea-vector are the coefficients of the polynomial expression, Equa. 30,for z′(x,y). It is to be noted that, although the polynomial expressionfor z′(x,y) used in this example is only of order 3 for ease ofillustration, the equations and methods of solution are easily extendedto higher order polynomials. Programs to solve the resulting equationscan be written using mathematical software available for use withpersonal computers. Of course, as the order of the polynomial increases,so too do the sizes of the M-matrix, and the b and a-vectors. Forinstance, associated with a polynominal of order 5 is a M-matrix having20×20 elements, and b and a-vectors each having 20 elements.$\begin{matrix}{M = \begin{bmatrix}{49} & {0} & {0} & 0 & 0 & {576} & 0 & {192} & 0 \\{0} & {49} & {0} & 0 & 0 & 0 & {192} & 0 & {576} \\{0} & {0} & {768} & 0 & 0 & 0 & 0 & 0 & 0 \\{0} & {0} & {0} & {384} & 0 & 0 & 0 & 0 & 0 \\{0} & {0} & {0} & 0 & {768} & 0 & 0 & 0 & 0 \\{576} & {0} & 0 & 0 & 0 & {14040} & 0 & {1380} & 0 \\{0} & {192} & 0 & 0 & 0 & 0 & {3400} & 0 & {1380} \\{192} & {0} & 0 & 0 & 0 & {1380} & 0 & {3400} & 0 \\{0} & {576} & 0 & 0 & 0 & 0 & {1380} & 0 & {14040}\end{bmatrix}} & (32) \\{b = {{\begin{bmatrix}{{- 0}{.02041}} \\{{- 0}{.001126}} \\{{- 49}{.09}} \\{0{.000007021}} \\{{- 48}{.69}} \\{{- 0}{.2415}} \\{0{.1101}} \\{{- 0}{.08047}} \\{0{.4015}}\end{bmatrix}\quad{and}\quad a} = \begin{bmatrix}{{- 0}{.0004123}} \\{{- 0}{.001039}} \\{{- 0}{.06392}} \\{0{.00000001828}} \\{{- 0}{.0634}} \\{{- 0}{.0000002603}} \\{0{.00006473}} \\{{- 0}{.0000002798}} \\{0{.00006487}}\end{bmatrix}}} & (33)\end{matrix}$

The expression for z′(x,y) defined by Equa. 30 is determined only up toan arbitrary constant. When one considers that a machining operationgenerally is employed to reshape surface z(x,y) to the new modifiedsurface z′(x,y) and that the machine has to be programmed to remove“positive” amounts of material, one realizes that the arbitrary constanthas to be large enough to shift surface z′(x,y) so that, when shifted,its value can never be greater than z(x,y). For the example, z′(x,y)values are shifted in the negative z-direction by 0.011247 mm to satisfythis condition. The depth of cut to modify z(x,y) to the correctingsurface is labelled CUT, and is expressed Equa. 34. Numerical values of1000*CUT (in microns) at 1 mm spacings in the x & y directions are givenin Equa. 35.

CUT(x,y)≡z(x, y)−(z′(x, y)−.011247)   (34) $\begin{matrix}{{1000 \cdot {CUT}} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 4 & {5{.5}} & {6{.3}} & {6{.3}} & {5{.6}} & 0 & 0 \\0 & 6 & {7{.9}} & {9{.3}} & {10} & {10{.1}} & {9{.6}} & {8{.5}} & {0} \\0 & {8{.1}} & {9{.7}} & {10{.9}} & {11{.5}} & {11{.7}} & {11{.4}} & {10{.6}} & {0} \\{6{.7}} & {8{.4}} & {9{.7}} & {10{.7}} & {11{.2}} & {11{.5}} & {11{.4}} & {10{.9}} & {10{.1}} \\0 & {7{.3}} & {8{.3}} & 9 & {9{.6}} & {9{.9}} & {9{.9}} & {9{.8}} & {0} \\0 & {5{.2}} & {5{.9}} & {6{.4}} & {6{.9}} & {7{.2}} & {7{.5}} & {7{.7}} & {0} \\0 & 0 & {2{.8}} & {3{.2}} & {3{.6}} & 4 & {4{.5}} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}} & (35)\end{matrix}$

4.2.5 Check that the New Surface Corrects Aberrations

In order to realize how well the new surface z′(x,y), described by thepolynomial expression (Equa. 30) with the a-coefficients (Equa. 33),corrects the original optical aberrations, the following calculationsare done: 1) find δz′/δx and δz/δy from Equa. 30 at all the measurementsites, and 2) use the values just found to find the x, y and zcoordinates of N″ from Equa. 1 & 2 where N″ represents the normal toz′(x,y) [see Equa. 30]. Next rewrite Equa. 11 in the following formwhich takes into account the labelling for the new correcting surface:B″≡n·A−λ″·N″  (36)

B″ represents the emerging rays after the surface has been corrected andis the parameter currently being sought. A, which is invariant withchanges to the optical surface, is already known having been obtainedpreviously by the methods described in the paragraph preceeding Equa.30. Likewise N″ is known, having been found by the methods outlined inthe paragraph just above. At this stage, λ″ is not yet determined;however, it is determined by the following methods.

First, from Equa. 9 find:β″=asin(|N″×B″|)   (37)Since the vector product N″×N″=0, from Equa. 36 find thatN″×B″=n*(N″×A). Therefore, Equa. 37 can be rewritten.β″=asin(|n*(N″×A)|)   (38)Since n, N″ and A are now known parameters, Equa. 38 gives the solutionfor β″.

Now that solutions for β″ are found, the corresponding solutions for α″and λ″ are found from modified forms of Equas. 10 & 12 which appear as .. .α″=asin(sin(β″)/n)   (39)λ″=n* cos(α″)−cos(β″)   (40)

B″ can now be found from Equa. 36 by substituting the now known valuesfor n, A, λ″ and N″. In Equas. 41 & 42, the x & y components of B″,which are written as B″x and B″y, appear multiplied by 1000 which makesthe resulting directional units in milliradians. $\begin{matrix}{{{1000 \cdot B^{''}}x} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {- 0.15} & 0.02 & 0 & {- 0.02} & 0.15 & 0 & 0 \\0 & {- 0.22} & 0.16 & 0.17 & 0 & {- 0.17} & {- 0.16} & 0.22 & 0 \\0 & 0.05 & 0.34 & 0.26 & 0 & {- 0.26} & {- 0.34} & {- 0.05} & 0 \\{- 0.67} & 0.14 & 0.4 & 0.29 & 0 & {- 0.3} & {- 0.41} & {- 0.14} & 0.67 \\0 & 0.05 & 0.34 & 0.26 & 0 & {- 0.27} & {- 0.34} & {- 0.05} & 0 \\0 & {- 0.23} & 0.16 & 0.17 & 0 & {- 0.17} & {- 0.16} & 0.23 & 0 \\0 & 0 & {- 0.15} & 0.02 & 0 & {- 0.02} & 0.16 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}} & (41) \\{{{1000 \cdot B^{''}}y} = \begin{bmatrix}0 & 0 & 0 & 0 & 0.63 & 0 & 0 & 0 & 0 \\0 & 0 & 0.22 & {- 0.04} & {- 0.13} & {- 0.05} & 0.21 & 0 & 0 \\0 & 0.15 & {- 0.15} & {- 0.33} & {- 0.39} & {- 0.33} & {- 0.15} & 0.14 & 0 \\0 & {- 0.01} & {- 0.16} & {- 0.26} & {- 0.29} & {- 0.26} & {- 0.17} & {- 0.01} & 0 \\0 & 0 & 0 & {- 0.01} & {- 0.01} & {- 0.01} & 0 & 0 & 0 \\0 & 0.02 & 0.16 & 0.25 & 0.28 & 0.25 & 0.16 & 0.02 & 0 \\0 & {- 0.14} & 0.15 & 0.33 & 0.39 & 0.33 & 0.16 & {- 0.14} & 0 \\0 & 0 & {- 0.22} & 0.05 & 0.14 & 0.05 & {- 0.21} & 0 & 0 \\0 & 0 & 0 & 0 & {- 0.65} & 0 & 0 & 0 & 0\end{bmatrix}} & (42)\end{matrix}$

A measure of the degree of collimation of all the B″-rays is itsroot-mean-square value, labeled B″rms. It is found by taking the squareroot of the average of all 49 values of B″x²+B″y². The value ofB″rms=0.00033 radians. That it is considerably less than the rms valuefor the original rays, which was Brms=0.00148 radians, demonstrates thesuccess of the algorithms and methods.

With polynomial expansions for z′(x,y) having orders higher than 3 as inthis current example, the corresponding size of B″rms is even less thanthe value given above. Furthermore, considering that 20/20 visual acuityimplies resolving lines spaced 0.00029 radian apart (i.e. 1 minute ofarc), the new correcting surface in this example is shown to improvevisual acuity to very nearly 20/20.

The centroids of the focal spots of the wavefront sensor pattern for thecase of the corrected B″-rays from the new correcting surface z′(x,y)are shown in FIG. 5. When compared to FIG. 4, which shows the patternfor the original aberrating optical surface, FIG. 5 shows rays that aremuch better collimated.

4.3 Fabrication of Aberration Correcting Surfaces on Lenses and CornealTissue

4.3.1 Diamond Point Machining

Since the surface contours of aberration-correcting contact lensesdescribed by z′(x,y) are more complex than the spherical or toroidalsurfaces of conventional contact lenses, the position of a cutting toolneeded to generate the z′(x,y) surfaces has to be controlled in a uniqueway. A programmable computer controlled single point diamond turning(SPDT) lathe, shown in FIG. 6, has been used to generate the surfaces ofaberration-correcting contact lenses. The lathe 30 has two movingsubassemblies: 1) a low vibration air bearing spindle 31, and 2) x-zpositioning slides 32. Contact lens 33, which is held and centered onthe end of spindle 31, is rotated by the spindle while a fine diamondtool 34 is moved by the positioning slides 32 with sub-micron resolutionboth perpendicular to (i.e. tool transverse scan in x-direction), andparallel to (i.e. tool cutting depth in z-direction) the direction ofthe spindle's axis. Smooth tool motion is achieved by the preferredmeans of using hydrostatic oil-bearing or air-bearing slideways. Apreferred means for precise slide positioning is achieved by usingcomputer controlled piezoelectric drivers or precise lead screw drivers(not shown). Since the machine must be completely free of both internaland external vibrations, both lathe 30 and x-z slides 32 are secured toa pneumatically isolated table top 35 which rests on granite base 36.

During lens machining, a computer 37 receives synchronous signals fromspindle 31 and controls the movement of the x-z translation slides 32along a programmed trajectory that is synchronized with the rotationalposition of the spindle. The motion of the x-translation slide (i.e.perpendicular to the spindle axis) generally is at a uniform speed aswith an ordinary lathe. However, the requirement of forming anon-axially symmetric lens surface (i.e. z′(x,y) shape), when using ahigh speed lathe, requires a unique control method for positioning thez-translation slide (i.e. controls cutter movement parallel to thespindle axis and, consequently, the depth of cut on the lens surfacewhich rotates rapidly with respect to the cutter). The z-translationslide must be rapidly and precisely located in accord with both thex-translation slide location, and the rotational position of thespindle. The preferred means of rapid and precise positioning of thez-translation slide is by the utilization of computer controlledpiezoelectric drivers.

Finally, mounting the optical element firmly on a supporting blockbefore placement on the end of the spindle is extremely important toavoid distorting the optical surface during machining. Care also must betaken to avoid distorting the lens blank when securing it to thesupporting block; otherwise, the surface will warp after removing itfrom the block following final surfacing.

As an alternative to a precision lathe, custom contact lenses can alsobe machined by an x-y-z contour cutting machine. With a machine of thistype, the lens is held stationary and figured by a cutting tool drivenin a precise x-y raster scan while the depth of cut in the z-directionis controlled by a computer often with positional feedback provided byan interferometer. Such an x-y-z contour cutting machine is described byPlummer et al. at the 8th International Precision Engineering Seminar(1995), and published in those proceedings by the Journal of AmericanSociety of Precision Engineering, pp. 24-29.

4.3.2 Surfacing by Ablation using an Ultraviolet Laser

Laser ablation of corneal tissue currently is used to reshape theanterior corneal surface in order to correct near and farsightedness andastigmatism in the human eye. Laser tissue ablation also can be used tomake more subtle corrections to the corneal surface than are nowperformed. Such corrections are needed to correct the eye's higher orderoptical aberrations. Therefore, laser surface ablation is a preferredmethod of contouring corneal tissue in order to correct the eye's higherorder optical aberrations.

Also, as an alternative to machining optical plastic contact lenses witha fine diamond point tool, laser machining, similar to the laserablation technique used to reshape the surface of the cornea, is alsopossible. It is likely that the most useful lasers for this purpose willprove to be those emitting ultra-violet light which ablate material awayfrom surfaces, not by thermal heating or melting, but by rapiddisruption of chemical bonds. Excimer lasers and solid statefrequency-tripled (or frequency-quadrupled) Nd-YAG lasers are nowproving to be useful both for machining plastics and ablating humantissue. These same lasers may be considered for precise surfacingoperations needed for the fabrication of aberration correcting contactlenses or advanced refractive surgery.

4.4 Alternate Methods for Fabrication of Aberration Correcting Lenses

In addition to diamond point machining and laser ablation methods, thereare other conceivable ways of fabricating an optical lens that cancompensate for the eye's irregular optical aberrations. Any successfulfabrication method must be capable of either precisely forming the lenssurface to the required z′(x,y) shape, or meticulously varying therefractive index over the surface in order to bend the light rays tocorrect the aberrations. Described below are several alternate lensfabrication methods.

4.4.1 Mechanical Force Thermal Molding

If the lens is a thermoplastic, its surface may be formed to match thatof a heated die. This method follows from the techniques used to formplastic parts by injection or compression molding. Since the desiredtopography of the lens surface generally is complex, the surface of theheated die is required to have a corresponding complex shape. Whennormally forming a plastic lens by the methods of injection orcompression molding, the die which determines the shape of the lens ismade of a single piece of metal and has a permanent surface shape. Toform the customized and complex lens surfaces needed to correct theirregular aberrations of the human eye, it is necessary to construct andutilize dies with variable or “adaptive” surfaces. In the field known as“adaptive optics”, such variable surfaces (anecdotally referred to as“rubber mirrors”) have been formed and controlled using computer-derivedinput signals that drive electromechanical fingers which press against adeformable metal surface.

Such an arrangement, which is called here a variable segmented die, isshown in FIG. 7. The housing for the die 40 provides a number ofchanneled holes to hold a corresponding number of mechanical fingers 41which are moved upwards or downwards in the die housing 40 by electronicdrivers or actuators 42. Mechanical fingers 41 press against acontinuous deformable metal surface 43 which contacts the thermoplasticlens surface and establishes its final shape. As an alternative to usinga customized heated die with its surface formed by electromechanicalfingers pressing in a controlled way against a deformable metal surface,an arrangement of close-packed, electronically actuated mechanicalfingers without a deformable metal surface may be used. With thisalternative arrangement, each mechanical finger in the array would pressdirectly on the surface of the thermoplastic lens blank in order to formits surface as required.

4.4.2 Photolithography and Etching

A layer of photoresist is spun onto the glass or plastic lens substrate,and selectively exposed using either an optical scanner or an electronbeam scanner. The exposure extent over regions of the surface of thephotoresist is properly matched with the desired surface contour of thefinished lens. The photoresist is developed chemically, thereby beingselectively prepared for subsequent etching over those areas havingpreviously received the most illumination. The surface of the glass orplastic lens substrate is then etched using reactive ion etching orchemically assisted ion-beam etching where the depth of the etch isdetermined by the extent of illumination during the previous exposure ofthe photoresist. It is noted that forms of PMMA (i.e. polymethylmethacrylate), which is a widely used optical plastic used for makingcontact lenses, are used as photoresists. Therefore, one can imaginetailoring the surface of a PMMA lens, in order to correct an eye'shigher-order aberrations, directly without needing to use a separatephotoresist layer over the lens substrate.

4.4.3 Thin Film Deposition:

There are various techniques now employed to add thin layers of variousmaterials to the surfaces of glass and plastic optical lenses. Thecurrent purposes for applying such thin films to lenses are: 1) to limitthe transmission of light, 2) to reduce surface reflections, and 3) toprotect surfaces from scratching and abrasion. Some of the methodsemployed include dip coating, spin coating, evaporative coating,spraying, and ion sputtering. To modify these existing methods for thepurpose of making a lens useful to correct the higher-order aberrationsof the human eye will require refinements that allow the deposition oflayers with thicknesses that vary selectively over the surface of thelens substrate in order to achieve the required z′(x,y) shape.

In addition to the methods now used to deposit thin films on opticalsurfaces, one can imagine other methods that may prove possibly moreuseful. Particularly useful could be a method of injecting from a finenozzle or series of fine nozzles, which are in proximity to a surface,controlled amounts of transparent liquid materials that permanently bondwhen deposited on the surface. Scanning these controllablematerial-depositing nozzles over the surface of an optical substrate canresult in building up on the optical substrate a custom surface contourthat meets the requirements of correcting the higher-order opticalaberrations of a subject's eye. Similar to what is conceived above isthe operation of ink-jet devices used in computer printers.

4.4.4 Surface Chemistry Alteration:

Surfaces of flat glass disks have been implanted with certain ionicimpurities that result in index of refraction changes that can varyradially from the center to the edge of the disk, or axially over thedepth of the disk. Such methods are utilized in the fabrication ofso-called “gradient index” optical elements, and flat disks areavailable commercially that behave as ordinary positive or negativelenses. Although plastic gradient index lenses have not yet been made,one can imagine altering the refractive index of a plastic subsurface bycertain techniques. For example, the exposure of a plastic surface toultraviolet light can alter the polymerization of subsurfacemacromolecules and, thereby, change the index of refraction in thesubsurface region. For another example, the methods now used to imbibedye molecules from solution into plastic surfaces in order to tintophthalmic lenses may also be effective in changing the index ofrefraction in subsurface layers. In the future, it is conceivable thatthe methods of surface chemistry alteration, which are used now tofabricate “gradient index” optical elements, can be refined for bothglass and plastic materials in order to make them useful in thefabrication of adaptive optical lenses for correcting the higher-orderaberrations of the human eye.

REFERENCES CITED

U.S. Patent Documents:

-   U.S. Pat. No. 5,777,719 issued Jul. 7, 1998 to David R. Williams and    Junzhong Liang entitled “Method and apparatus for improving vision    and the resolution of retinal images”

Other References:

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1. A method for correcting the optical aberrations beyond defocus andastigmatism of an eye fitted with an original contact lens having aknown anterior surface shape by providing a modified or new contact lenswhich has its anterior surface reshaped from said original contactlens's anterior surface, comprising the steps of: a) measuring saidoptical aberrations of an eye fitted with an original contact lens, b)performing a mathematical analysis of said eye's optical aberrationswhen fitted with original contact lens to determine said modifiedanterior contact lens surface shape, and c) fabricating said modifiedanterior contact lens surface by methods that remove, add or compressmaterial or alter the surface chemistry.
 2. A method as claimed in claim1 wherein said measuring of the eye's optical aberrations comprises thesub-steps of: i) optically projecting the image of a small point ofincoherent light onto the macular region of the eye's retina, ii)optically conveying the image of the eye's pupil, through which lightscattered back from the macular region emerges, onto a microlens array,iii) optically conveying the multiple spot images formed by saidmicrolens array onto the image plane of a photo-electronic imagingdevice, iv) transforming by means of the photo-electronic imaging devicethe multiple spot images formed by said microlens array to an electronicsignal which represents the images, v) conveying said electronic signalto a computer for data processing, vi) processing first the electronicsignal with said computer in order to obtain the coordinate locations ofthe centroids of said multiple spot images formed by said microlensarray, and vii) processing next said coordinate locations with saidcomputer in order to obtain the slopes of optical rays emerging from thesubject's pupil at said coordinate locations.
 3. A method as claimed inclaim 1 wherein said mathematical analysis comprises the sub-steps of:i) determining mathematically the normal vectors of said originalcontact lens's anterior surface, ii) determining mathematically thedirectional derivatives of said modified or new contact lens's anteriorsurface using data of said normal vectors of original contact lens'santerior surface and data of said eye's optical aberrations, and iii)fitting mathematically by the method of least squares said directionalderivatives to the corresponding directional derivatives of a polynomialexpression that represents said modified or new contact lens's anteriorsurface.
 4. A method as claimed in claim 1 wherein said step offabricating said modified or new contact lens's anterior surface ischosen from the group of methods comprising diamond point machining,laser ablation, thermal molding, photo-lithographic etching, thin filmdeposition, and surface chemistry alteration.
 5. A method for correctingthe optical aberrations beyond defocus and astigmatism of an eye with anoriginal anterior corneal surface of known shape by providing a modifiedanterior corneal surface shape, comprising the steps of: a) measuringsaid eye's optical aberrations, b) performing a mathematical analysis ofsaid eye's optical aberrations to determine said modified anteriorcorneal surface shape, c) fabricating said modified anterior cornealsurface by laser ablation.
 6. A method as claimed in claim 5 whereinsaid measuring of the eye's optical aberrations comprises the sub-stepsof: i) optically projecting the image of a small point of incoherentlight onto the macular region of the eye's retina, ii) opticallyconveying the image of the eye's pupil, through which light scatteredback from the macular region emerges, onto a microlens array, iii)optically conveying the multiple spot images formed by said microlensarray onto the image plane of a photo-electronic imaging device, iv)transforming by means of the photo-electronic imaging device themultiple spot images formed by said microlens array to an electronicsignal which represents the images, v) conveying said electronic signalto a computer for data processmg, vi) processing first the electronicsignal with said computer in order to obtain the coordinate locations ofthe centroids of said multiple spot images formed by said microlensarray, and vii) processing next said coordinate locations with saidcomputer in order to obtain the slopes of optical rays emerging from thesubject's pupil at said coordinate locations.
 7. A method as claimed inclaim 5 wherein said mathematical analysis comprises the sub-steps of:i) determining mathematically the normal vectors of said originalanterior corneal surface, ii) determining mathematically the directionalderivatives of said modified anterior corneal surface using data of saidnormal vectors of original anterior corneal surface and data of saideye's optical aberrations, and iii) fitting mathematically by the methodof least squares said directional derivatives to the correspondingdirectional derivatives of a polynomial expression that represents saidmodified anterior corneal surface.
 8. An ophthalmic device for measuringthe eye's optical aberrations either with or without a contact lens inplace on the cornea, including; a) an optical projection system forimaging a small point of light onto the macular region of the eye'sretina with an improvement provided by use of an incoherent light sourcechosen from the group comprising laser diodes operated below threshold,light emitting diodes, arc and plasma sources, and incandescent filamentlamps, b) an optical image acquisition system for conveying the image ofthe eye's pupil, through which light scattered back from the macularregion emerges, onto a microlens array, c) a microlens array to formmultiple spot images onto the image plane of a photo-electronic imagingdevice, d) a photo-electronic imaging device for transforming saidmultiple spot images formed by said microlens array to an electronicsignal which represents the images, e) a computer for processing theelectronic signal in order, first, to obtain the coordinate locations ofthe centroids of said multiple spot images formed by said microlensarray and, second, to obtain the slopes of optical rays emerging fromthe subject's pupil at said coordinate locations, and f) an opticalalignment system allowing the entering beam to be accurately centeredwith respect to the subject's pupil.
 9. An ophthalmic device as claimedin claim 8 wherein said optical projection system includes an opticalisolator consisting of a quarter-wave plate and polarizer.
 10. Anophthalmic device as claimed in claim 8 wherein said optical projectionsystem includes a field stop placed at a location that is opticallyconjugate to the eye's retina.
 11. An ophthalmic device as claimed inclaim 8 wherein said optical projection system includes both an opticalisolator consisting of a quarter-wave plate and polarizer, and a fieldstop placed at a location that is optically conjugate to the eye'sretina.
 12. An ophthalmic device as claimed in claim 8 wherein saidphoto-electronic imaging device is chosen from the group comprisingvidicons, charge-coupled devices, and charge-injection devices.
 13. Adevice for thermally forming surfaces on thermoplastic contact lensblanks that correct eyes' optical aberrations beyond defocus andastigmatism consisting of a die with an adjustable surface shape (eithercontinuous or discontinuous) formed by computer-controlledelectromechanical actuators or electromechanical fingers which are knownin the field of adaptive optics.
 14. A lathe device for machiningsurfaces on contact lens blanks that correct eyes' optical aberrationsbeyond defocus and astigmatism consisting of a rotating spindle ontowhich a contact lens blank is fastened, translation slides for preciselypositioning a diamond point cutting tool with respect to the surface ofthe contact lens blank, and a programmed computer that controls themovement of the translation slides synchronously with the rotationallocation of the spindle.
 15. A contour cutting device for machiningsurfaces on contact lens blanks that correct eyes'optical aberrationsbeyond defocus and astigmatism consisting of a means for supporting andholding stationary a contact lens blank, translation slides forprecisely positioning in three dimensions a diamond point cutting toolwith respect to the surface of the contact lens blank, and a programmedcomputer that controls the movement of the translation slides.
 16. Amethod for correcting optical aberrations beyond defocus and astigmatismof an eye comprising: a) fitting the eye with a first contact lenshaving a known anterior surface shape that is corrected for at leastfocus, b) measuring the optical aberrations of the eye fitted with thefirst contact lens, c) performing a mathematical analysis of the eye'soptical aberrations when fitted with the first contact lens to determinea modified anterior contact lens surface shape, and d) fabricating asecond contact lens having the modified anterior contact lens surface.17. A method as claimed in claim 16 performing the mathematical analysisto at least the eye's 4 ^(th) order optical aberrations of the eye whenfitted with the first contact lens to determine the second contact lenssurface shape, and the anterior surface of the second contact lens tohave the modified anterior surface that corrects the eye to at least the4 ^(th) order aberration.
 18. A method as claimed in claim 17 whereinthe measuring of the eye's optical aberrations comprises the sub-stepsof: i) optically projecting the image of a small point of incoherentlight onto the macular region of the eye's retina, ii) opticallyconveying the image of the eye's pupil, through which light scatteredback from the macular region emerges, onto a microlens array, iii)optically conveying the multiple spot images formed by the microlensarray onto the image plane of a photo-electronic imaging device, iv)transforming by means of the photo-electronic imaging device themultiple spot images formed by the microlens array to an electronicsignal which represents the images, v) conveying the electronic signalto a computer for data processing, vi) processing first the electronicsignal with the computer in order to obtain the coordinate locations ofthe centroids of the multiple spot images formed by the microlens array,and vii) processing next the coordinate locations with the computer inorder to obtain the slopes of optical rays emerging from the subject'spupil at the coordinate locations.
 19. A method as claimed in claim 17wherein the mathematical analysis comprises the sub-steps of: i)determining mathematically the normal vectors of the first contactlens's anterior surface, ii) determining mathematically the directionalderivatives of the second contact lens's anterior surface using data ofthe normal vectors of the first contact lens's surface and data of theeye's optical aberrations, and iii) fitting mathematically by the methodof least squares the directional derivatives to the correspondingdirectional derivatives of a polynomial expression that represents thesecond contact lens's anterior surface.
 20. A method as claimed in claim1 wherein the original contact lens's anterior surface contour functionz (x,y) and the eye's optical aberrations, represented by optical raysemerging from the pupil given by vector function B (x,y), are used tofind the surface contour function z′ (x,y) of the modified or newcontact lens by the following mathematical procedures: a) z′(x,y) isapproximated by the sum of a series of linearly independent terms in x &y with each term labeled by an index j wherein each term, a _(j) ·g_(j)(x,y), consists of an unknown constant coefficient a _(j) and aknown function g _(j)(x,y) as shown in following Equation ( 1 )$\begin{matrix}{{z^{\prime}\left( {x,y} \right)} \equiv {\sum\limits_{j}\quad{a_{j} \cdot {g_{j}\left( {x,y} \right)}}}} & (1)\end{matrix}$ b) following Equation ( 2 ) is obtained by taking vectorgradients of both sides of Equation ( 1 ) where grad z′ (x,y) is atwo-dimensional vector having components[δ  z^(′)  (x, y)/δ  x, δ  z^(′)  (x, y)/δ  y] $\begin{matrix}{{{grad}\quad{z^{\prime}\left( {x,y} \right)}} = {\sum\limits_{j}\quad{{a_{j} \cdot {grad}}\quad{g_{j}\left( {x,y} \right)}}}} & (2)\end{matrix}$ c) the normal vectors of the original contact lens'santerior surface, given by vector function N(x,y), are found by takingthe vector gradient of z(x,y) and normalizing it to unity as shown inthe following Equations ( 3A) to ( 3D) $\begin{matrix}{{{Nx} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad x}}},} & \left( {3A} \right) \\{{{Ny} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad y}}},} & \left( {3B} \right) \\{{Nz} \equiv \frac{1}{MAG}} & \left( {3C} \right)\end{matrix}$  MAG≡[1+(δz(x,y)/δx)²+(δz(x,y)/δy)²] ^(1/2)   ( 3D) d) therays incident on the original contact lens coming from within the eyeare given by vector function A(x,y) which is found by applying Snell'slaw of refraction at the air/lens interface which relates A(x,y) toknown vector function B(x,y) representing the emerging rays, and vectorfunction N(x,y) given by Equations ( 3A)-( 3D), e) the normal vectors ofthe modified or new contact len's anterior surface, given by the vectorfunction N′, are found from the following Equation ( 4 ) where vectorfunction B′ is represented by unit vectors pointed along the positivez-axis, and n is the lens's refractive index $\begin{matrix}{N^{\prime} \equiv \frac{\left( {{n \cdot A} - B^{\prime}} \right)}{{{n \cdot A} - B^{\backprime}}}} & (4)\end{matrix}$ f) the directional derivatives of the modified or newcontact lens's anterior surface are obtained from the following Equation( 5 ) using the components of vector function N′=[N′ _(x) , N′ _(y) , N′_(z) ] found from Equation ( 4 ) $\begin{matrix}{\frac{\delta\quad z^{\prime}}{\delta\quad x} \equiv {{- \frac{N_{x}^{\backprime}}{N_{z}^{\backprime}}}\quad{and}\quad\frac{\delta\quad z^{\prime}}{\delta\quad y}} \equiv {- \frac{N_{y}^{\backprime}}{N_{z}^{\backprime}}}} & (5)\end{matrix}$ g) apply the method of least squares to minimize thesquare the difference between the component values of grad z′(x,y) foundfrom Equation ( 5 ) and the component values of grad z′(x,y) given bythe approximation series, Equa. ( 2 ), in order to obtain matrixEquation ( 6 )M·a≡b   ( 6 ) where $\begin{matrix}{M_{({i,j})} \equiv {\sum\limits_{x,y}\quad{\left( {{grad}\quad{g_{i} \cdot {grad}}\quad g_{j}} \right)\quad{and}\quad b_{i}}} \equiv {\sum\limits_{x,y}\quad\left( {{grad}\quad{g_{i} \cdot {grad}}\quad z^{\backprime}} \right)}} & (7)\end{matrix}$ h) obtain the inverse of matrix M, and then find thea-coefficients from matrix Equation ( 8 )a≡M ⁻¹ ·b   ( 8 ) i) the a-coefficients determined from Equation ( 8 )are used in Equation ( 1 ) which defines the modified or new contactlens's anterior surface, z′ (x,y) when it is represented, for example,by a 5 ^(th) order Taylor series.
 21. A methdo as claimed in claim 19wherein the first Contact lens's anterior surface contour functionz(x,y) and the eye's optical aberrations, represented by optical raysemerging from the pupil given by vector function B(x,y), are used tofind the surface contour function z′(x−y) of the second contact lens bythe following mathematical procedures: a) z′ (x,y) is approximated bythe sum of a series of linearly independent terms in x & y with eachterm labeled by an index j wherein each term, a _(j) ·g _(j)(x,y),consists of an unknown constant coefficient a _(j) and a known functiong _(j)(x,y) as shown in following Equation ( 1 ) $\begin{matrix}{{z^{\prime}\quad\left( {x,y} \right)} \equiv {\sum\limits_{j}\quad{a_{j} \cdot {g_{j}\left( {x,y} \right)}}}} & (1)\end{matrix}$ b) following Equation ( 2 ) is obtained by taking vectorgradients of both sides of Equation ( 1 ) where grad z′ (x,y) is atwo-dimensional vector having components[δ  z^(′)  (x, y)/δ  x, δ  z^(′)  (x, y)/δ  y] $\begin{matrix}{{{grad}\quad{z^{\prime}\left( {x,y} \right)}} = {\sum\limits_{j}\quad{{a_{j} \cdot {grad}}\quad{g_{j}\left( {x,y} \right)}}}} & (2)\end{matrix}$ c) the normal vectors of the first contact lens's anteriorsurface, given by vector function N(x,y), are found by taking the vectorgradient of z(x,y) and normalizing it to unity as shown in the followingEquations ( 3A) to ( 3D) $\begin{matrix}{{{Nx} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad x}}},} & \left( {3A} \right) \\{{{Ny} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad y}}},} & \left( {3B} \right) \\{{Nz} \equiv \frac{1}{MAG}} & \left( {3C} \right)\end{matrix}$  MAG≡[1+(δz(x,y)/δx)²+(δz(x,y)/δy)²] ^(1/2)   ( 3D) d) therays incident on the first contact lens coming from within the eye aregiven by vector function A(x,y) which is found by applying Snell's lawof refraction at the air/lens interface which relates A(x,y) to knownvector function B(x,y) representing the emerging rays, and vectorfunction N(x,y) given by Equations ( 3A)-( 3B), e) the normal vectors ofthe second contact lens's anterior surface, given by the vector functionN′, are found from the following Equation ( 4 ) where vector function B′is represented by unit vectors pointing along the positive z-axis, and nis the lens's refractive index $\begin{matrix}{N^{\prime} = \frac{\left( {{n \cdot A} - B^{\prime}} \right)}{{{n \cdot A} - B^{\backprime}}}} & (4)\end{matrix}$ f) the directional derivatives of the second contactlens's anterior surface are obtained from the following Equation ( 5 )using the components of vector function N′=[N′ _(x) ,N′ _(y) ,N′ _(z) ]found from Equation ( 4 ) $\begin{matrix}{\frac{\delta\quad z^{\prime}}{\delta\quad x} \equiv {{- \frac{N_{x}^{\backprime}}{N_{z}^{\backprime}}}\quad{and}\quad\frac{\delta\quad z^{\prime}}{\delta\quad y}} \equiv {- \frac{N_{y}^{\backprime}}{N_{z}^{\backprime}}}} & (5)\end{matrix}$ g) apply the method of at least squares to minimize thesquare the difference between the component values of grad z′ (x,y)found from Equation ( 5 ) and the component values of grad z′ (x,y)given by the approximation series, Equa. ( 2 ), in order to obtainmatrix Equation ( 6 )M·a≡b   ( 6 ) where $\begin{matrix}{{M_{({i,j})} \equiv {\sum\limits_{x,y}\quad{\left( {{grad}\quad{g_{i} \cdot {grad}}\quad g_{j}} \right)\quad{and}\quad b_{i}}}} = {\sum\limits_{x,y}\quad\left( {{grad}\quad{g_{i} \cdot {grad}}\quad z^{\backprime}} \right)}} & (7)\end{matrix}$ h) obtain the inverse of matrix M, and then find thea-coefficient from matrix Equation ( 8 )a≡M ⁻¹ ·b   (8) i) the a-coefficients determined from Equation ( 8 ) areused in Equation ( 1 ) which defines the second contact lens's anteriorsurface, z′ (x,y) when it is represented, for example, by a 5 ^(th)order Taylor series.
 22. A method as claimed in claim 5 wherein theoriginal cornea's anterior surface contour function z(x,y) and the eye'soptical aberrations, represented by optical rays emerging from the pupilgiven by vector function B(x,y), are used to find the surface contourfunction z′(x,y) of the modified cornea by the following mathematicalprocedures: a) z′ (x,y) is approximated by the sum of a series oflinearly independent terms in x & y with each term labeled by an index jwherein each term, a _(j) ·g _(j)(x,y), consists of an unknown constantcoefficient a _(j) and a known function g _(j)(x,y) as shown infollowing Equation ( 1 ) $\begin{matrix}{{z^{\prime}\quad\left( {x,y} \right)} \equiv {\sum\limits_{j}\quad{a_{j} \cdot {g_{j}\left( {x,y} \right)}}}} & (1)\end{matrix}$ b) following Equation ( 2 ) is obtained by taking vectorgradients of both sides of Equation ( 1 ) where grad z′ (x,y) is atwo-dimensional vector having components[δ  z^(′)  (x, y)/δ  x, δ  z^(′)  (x, y)/δ  y] $\begin{matrix}{{{grad}\quad{z^{\prime}\left( {x,y} \right)}} = {\sum\limits_{j}\quad{{a_{j} \cdot {grad}}\quad{g_{j}\left( {x,y} \right)}}}} & (2)\end{matrix}$ c) the normal vectors of the original cornea's anteriorsurface, given by vector function N(x,y), are found by taking the vectorgradient of z(x,y) and normalizing it to unity as shown in the followingEquations ( 3A) to ( 3D) $\begin{matrix}{{{Nx} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad x}}},} & \left( {3A} \right) \\{{{Ny} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad y}}},} & \left( {3B} \right) \\{{Nz} \equiv \frac{1}{MAG}} & \left( {3C} \right)\end{matrix}$  MAG≡[1+(δz(x,y)/δx)²+(δz(x,y)/δy)²] ^(1/2)   ( 30 ) d)the rays incident on the original cornea coming from within the eye aregiven by vector function A(x,y) which is found by applying Snell's lawof reflection at the air/cornea interface which relates A(x,y) to knownvector function B(x,y) representing the emerging rays, and vectorfunction N(x,y) given by Equations ( 3A)-( 3B), e) the normal vectors ofthe modified cornea's anterior surface, given by the vector function N′,are found from the following Equation ( 4 ) where vector function B′ isrepresented by unit vectors pointing along the positive z-axis, and n isthe cornea's refractive index $\begin{matrix}{N^{\prime} \equiv \frac{\left( {{n \cdot A} - B^{\prime}} \right)}{{{n \cdot A} - B^{\backprime}}}} & (4)\end{matrix}$ f) the directional derivatives of the modified cornea'santerior surface are obtained from the following Equation ( 5 ) usingthe components of vector function N′=[N′ _(x) , N′ _(y) , N′ _(z) ]found from Equation ( 4 ) $\begin{matrix}{\frac{\delta\quad z^{\prime}}{\delta\quad x} \equiv {{- \frac{N_{x}^{\backprime}}{N_{z}^{\backprime}}}\quad{and}\quad\frac{\delta\quad z^{\prime}}{\delta\quad y}} \equiv {- \frac{N_{y}^{\backprime}}{N_{z}^{\backprime}}}} & (5)\end{matrix}$ g) apply the method of least squares to minimize thesquare the difference between the component values of grad z′(x,y) foundthe Equation ( 5 ) and the component values of grad z′ (x,y) given bythe approximation series, Equa. ( 2 ), in order to obtain matrixEquation ( 6 )M·a≡b   ( 6 ) where $\begin{matrix}{{M_{({i,j})} \equiv {\sum\limits_{x,y}{\left( {{grad}\quad{g_{i} \cdot {grad}}\quad g_{j}} \right)\quad{and}\quad b_{i}}}} = {\sum\limits_{x,y}\left( {{grad}\quad{g_{i} \cdot {grad}}\quad z^{\backprime}} \right)}} & (7)\end{matrix}$ h) obtain the inverse of matrix M, and then find thea-coefficients from matrix Equation ( 8 )a≡M ⁻¹ ·b i) the a-coefficient determined from Equation ( 8 ) are usedin Equation ( 1 ) which defines the modified cornea's anterior surface,z′ (x,y) when it is represented, for example, by a 5 ^(th) order Taylorseries.
 23. A contact lens comprising an anterior surface which isfabricated to correct the optical aberration to at least 4 ^(th) orderof a person's eye.
 24. The contact lens of claim 23 wherein themeasuring of the eye's optical aberrations is by steps of: (i) opticallyprojecting the image of a small point of incoherent light onto themacular region of the eye's retina, (ii) optically conveying the imageof the eye's pupil, through which light scattered back from the macularregion emerges onto a microlens array, (iii) optically conveying themultiple spot images formed by the microlens array onto the image planeof a photo-electronic imaging device, (iv) transforming by means of thephoto-electronic imaging device the multiple spot images formed by themicrolens array to an electronic signal which represents the images; (v)conveying the electronic signal to a computer for data processing, (vi)processing first the electronic signal with the computer in order toobtain the coordinate locations of the centroids of the multiple spotimages formed by the microlens array, and (vii) processing next thecoordinate locations with the computer in order to obtain the slopes ofoptical rays emerging from the subject's pupil at the coordinatelocations.
 25. The contact lens of claim 23 wherein the mathematicalanalysis comprises the sub-steps of: (a) determining mathematically thenormal vectors of a first contact lens anterior surface; (b) determiningmathematically the directional derivatives of the a second contact lensanterior surface using data of the normal vectors of the first contactlens anterior surface and data of the eye's optical aberrations; and (c)fitting mathematically by the method of least squares the directionalderivatives to the corresponding directional derivatives of a polynomialexpression that represents the second contact lens anterior surface. 26.The contact lens of claim 23 wherein the first contact lens anteriorsurface contour function z(x,y) and the eye's optical aberrations,represented by optical rays emerging from the pupil given by vectorfunction B(x,y), are used to find the surface contour function z′(x,y)of the second contact lens by the following mathematical procedures: (a)z′(x,y) is approximated by the sum of a series of linearly independentterms in x & y with each term labeled by an index j wherein each term, a_(j) ·g _(j)(x,y), consists of an unknown constant coefficient a _(j)and a known function g _(j) (x,y) as shown in following Equation ( 1 )$\begin{matrix}{{z^{\prime}\left( {x,y} \right)} \equiv {\sum\limits_{j}{a_{j} \cdot {g_{j\quad}\left( {x,y} \right)}}}} & (1)\end{matrix}$ b) following Equation ( 2 ) is obtained by taking vectorgradients of both sides of Equation ( 1 ) where grad z′ (x,y) is atwo-dimensional vector having components [δ z′(x,y)/δx, δz′(x,y)/δy]$\begin{matrix}{{{grad}\quad{z^{\prime}\left( {x,y} \right)}} = {\sum\limits_{j}{{a_{j} \cdot {grad}}\quad{g_{j}\left( {x,\quad y} \right)}}}} & (2)\end{matrix}$ c) the normal vectors of the first contact lens's anteriorsurface, given by vector function N (x,y), are found by taking thevector gradient of z(x,y) and normalizing it to unity as shown in thefollowing Equations ( 3A) to ( 3D) $\begin{matrix}{{{Nx} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad x}}},} & \left( {3A} \right) \\{{{Ny} \equiv {\frac{1}{MAG} \cdot \frac{{- \delta}\quad{z\left( {x,y} \right)}}{\delta\quad y}}},} & \left( {3B} \right) \\{{Nz} \equiv \frac{1}{MAG}} & \left( {3C} \right)\end{matrix}$  MAG≡[1+(δz(x,y)/δx)²+(δz(x,y)/δy)²]^(½)  ( 3D) d) therays incident on the first contact lens coming from within the eye aregiven by vector function A(x,y) which is found by applying Snell's lawof refraction at the air/lens interface which relates A(x,y) to knownvector function B(x,y) representing the emerging rays, and vectorfunction N(x,y) given by Equations ( 3A)−( 3B), e) the normal vectors ofthe second contact lens's anterior surface, given by the vector functionN′, are found from the following Equation ( 4 ) where vector function B′is represented by unit vectors pointing along the positive z-axis, and nis the lens's (or cornea's) refractive index $\begin{matrix}{N^{\prime} \equiv \frac{\left( {{n \cdot A} - B^{\prime}} \right)}{{{n \cdot A} - B^{\backprime}}}} & (4)\end{matrix}$ f) the directional derivatives of the second contactlens's anterior surface are obtained from the following Equation ( 5 )using the components of vector function N′=[N′ _(x) , N′ _(y) , N′ _(z)] found from Equation ( 4 ) $\begin{matrix}{\frac{\delta\quad z^{\prime}}{\delta\quad x} \equiv {{- \frac{N_{x}^{\backprime}}{N_{z}^{\backprime}}}\quad{and}\quad\frac{\delta\quad z^{\prime}}{\delta\quad y}} \equiv {- \frac{N_{y}^{\backprime}}{N_{z}^{\backprime}}}} & (5)\end{matrix}$ g) apply the method of least squares to minimize thesquare the difference between the component values of grad z′ (x,y)found from Equation ( 5 ) and the component values of grad z′(x,y) givenby the approximation series, Equa. ( 2 ), in order to obtain matrixEquation ( 6 )M·a≡b   ( 6 ) where $\begin{matrix}{M_{({i,j})} \equiv {\sum\limits_{x,y}{\left( {{grad}\quad{g_{i} \cdot {grad}}\quad g_{j}} \right)\quad{and}\quad b_{i}}} \equiv {\sum\limits_{x,y}\left( {{grad}\quad{g_{i} \cdot {grad}}\quad z^{\backprime}} \right)}} & (7)\end{matrix}$ h) obtain the inverse of matrix M, and then find thea-coefficients from matrix Equation ( 8 )a≡M ⁻ ·b   ( 8 ) i) the a-coefficients determined from Equation ( 8 )are used in Equation ( 1 ) which defines the second contact lens'santerior surface, z′ (x,y) when it is represented, for example, by a 5^(th) order Taylor series.